DOCSLIB.ORG

Computational Tools for Topological Cohochschild Homology 3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computational Tools for Topological Cohochschild Homology 3 COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY ANNA MARIE BOHMANN, TEENA GERHARDT, AMALIE HØGENHAVEN, BROOKE SHIPLEY, AND STEPHANIE ZIEGENHAGEN Abstract. In recent work, Hess and Shipley [18] defined a theory of topo- logical coHochschild homology (coTHH) for coalgebras. In this paper we de- velop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coB¨okstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations. 1. Introduction The theory of Hochschild homology for algebras has a topological analogue, called topological Hochschild homology (THH). Topological Hochschild homology for ring spectra is defined by changing the ground ring in the ordinary Hochschild complex for rings from the integers to the sphere spectrum. For coalgebras, there is a theory dual to Hochschild homology called coHochschild homology. Variations of coHochschild homology for classical coalgebras, or corings, appear in [7, Section 30], and [11], for instance, and the coHochschild complex for differential graded coalgebras appears in [17]. In recent work [18], Hess and Shipley define a topological version of coHochschild homology (coTHH), which is dual to topological Hochschild homology. In this paper we develop computational tools for coTHH. We begin by giving a very general definition for coTHH in a model category endowed with a symmetric monoidal structure in terms of the homotopy limit of a cosimplicial object. This level of generality makes concrete calculations difficult; nonetheless, we give more accessible descriptions of coTHH in an assortment of algebraic contexts. One of the starting points in understanding Hochschild homology is the Hoch- arXiv:1706.01908v1 [math.AT] 6 Jun 2017 schild–Kostant–Rosenberg theorem, which, in its most basic form, identifies the Hochschild homology of free commutative algebras. See [20] for the classical result and [29] for an analogue in the category of spectra. The set up for coalgebras is not as straightforward as in the algebra case and we return to the general situation to analyze the ingredients necessary for defining an appropriate notion of cofree coalgebras. In Theorem 3.11 we conclude that the Hochschild–Kostant–Rosenberg theorem for cofree coalgebras in an arbitrary model category boils down to an anal- ysis of the interplay between the notion of “cofree” and homotopy limits. We then Date: June 8, 2017. 2010 Mathematics Subject Classification. Primary: 19D55, 16T15; Secondary: 16E40, 55U35, 55P43 . Key words and phrases. Topological Hochschild homology, coalgebra, Hochschild–Kostant– Rosenberg. 1 2 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN prove our first computational result, a Hochschild–Kostant–Rosenberg theorem for cofree differential graded coalgebras. A similar result has been obtained by Farinati and Solotar in [13] and [14] in the ungraded setting. Theorem 1.1. Let X be a nonnegatively graded cochain complex over a field k. Let Sc(X) be the cofree coaugmented cocommutative coassociative conilpotent coalgebra over k cogenerated by X. Then there is a quasi-isomorphism of differential graded k-modules c coTHH(Sc(X)) ≃ ΩS (X)|k. where the right hand side is an explicit differential graded k-module defined in Sec- tion 3. c The precise definition of ΩS (X)|k depends on the characteristic of k. Let U(Sc(X)) denote the underlying differential graded k-modules of the cofree coalgebra Sc(X). If char(k) 6= 2, we can compute coTHH(Sc(X)) as U(Sc(X))⊗U(Sc(Σ−1X)), while in characteristic 2, we have coTHH(Sc(X)) =∼ U(Sc(X)) ⊗ Λ(Σ−1X), where Λ de- notes exterior powers. Definitions of all of the terms here appear in Sections 2 and 3 and this theorem is proved as Theorem 3.15. We then develop calculational tools to study coTHH(C) for a coalgebra spectrum C in a symmetric monoidal model category of spectra; see Section 4. Recall that for topological Hochschild homology, the B¨okstedt spectral sequence is an essential computational tool. For a field k and a ring spectrum R, the B¨okstedt spectral sequence is of the form 2 E∗,∗ = HH∗(H∗(R; k)) ⇒ H∗(THH(R); k). This spectral sequence arises from the skeletal filtration of the simplicial spectrum THH(R)•. Analogously, for a coalgebra spectrum C, we consider the Bousfield– Kan spectral sequence arising from the cosimplicial spectrum coTHH•(C). We call this the coB¨okstedt spectral sequence. We identify the E2-term in this spectral sequence and see that, as in the THH case, the E2-term is given by a classical algebraic invariant: Theorem 1.2. Let k be a field. Let C be a coalgebra spectrum that is cofibrant as an underlying spectrum. The Bousfield–Kan spectral sequence for coTHH(C) gives a coB¨okstedt spectral sequence with E2-page s,t k E2 = coHHs,t(H∗(C; k)), that abuts to Ht−s(coTHH(C); k). As one would expect with a Bousfield–Kan spectral sequence, the coB¨okstedt spectral sequence does not always converge. However, we identify conditions under which it converges completely. In particular, if the coalgebra spectrum C is a ∞ suspension spectrum of a simply connected space X, denoted Σ+ X, the coB¨okstedt ∞ spectral sequence converges completely to H∗(coTHH(Σ+ X); k) if a Mittag-Leffler condition is satisfied. Further, we prove that for C connected and cocommutative this is a spectral sequence of coalgebras. Using this additional algebraic structure we prove several computational results, including the following. The divided power coalgebra Γk[−] and the exterior coalgebra Λk(−) are introduced in detail in Section 5. Note that if X is a graded k-vector space concentrated in even degrees and with basis (xi)i∈N, COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 3 c Sc(X)|k there are identifications S (X) =∼ Γk[x1, x2,... ] and Ω =∼ U(Γk[x1, x2,... ])⊗ U(Λk[z1,z2,... ]). We improve Theorem 1.1 in Proposition 5.1 by showing that under the above conditions on X there is indeed an isomorphism of coalgebras c coHH(S (X)) =∼ Γk[x1, x2,... ]) ⊗ Λk[z1,z2,... ]. Hence Theorem 1.2 and the coalgebra structure on the spectral sequence yield the following result. Theorem 1.3. Let C be a cocommutative coassociative coalgebra spectrum that is cofibrant as an underlying spectrum, and whose homology coalgebra is H∗(C; k)=Γk[x1, x2,... ], where the xi are cogenerators in nonnegative even degrees, and there are only finitely many cogenerators in each degree. Then the coB¨okstedt spectral sequence for C collapses at E2, and E2 =∼ E∞ =∼ Γk[x1, x2,... ] ⊗ Λk(z1,z2,... ), with xi in degree (0, deg(xi)) and zi in degree (1, deg(xi)). This theorem should thus be thought of as a computational analogue of the Hochschild–Kostant–Rosenberg Theorem at the level of homology. These computational results apply to determine the homology of coTHH of many suspension spectra, which have a coalgebra structure induced by the diagonal map ∞ on spaces. For simply connected X, topological coHochschild homology of Σ+ X ∞ coincides with topological Hochschild homology of Σ+ ΩX, and thus is closely re- lated to algebraic K-theory, free loop spaces, and string topology. Our results give a new way of approaching these invariants, as we briefly recall in Section 5. The paper is organized as follows. In Section 2, we define coTHH for a coalge- bra in any model category endowed with a symmetric monoidal structure as the homotopy limit of the cosimplicial object given by the cyclic cobar construction coTHH•. We then give conditions for when this homotopy limit can be calculated efficiently and explain a variety of examples. In Section 3, we develop cofree coalgebra functors. These are not simply dual to free algebra functors, and we are explicit about the conditions on a symmetric monoidal category that are necessary to make these functors well behaved. In this context an identification similar to the Hochschild–Kostant–Rosenberg Theorem holds on the cosimplicial level. We also prove a dg-version of the Hochschild– Kostant–Rosenberg theorem. In Section 4 we define and study the coB¨okstedt spectral sequence for a coalgebra spectrum. In particular, we analyze the convergence of the spectral sequence, and prove that it is a spectral sequence of coalgebras. In Section 5 we use the coB¨okstedt spectral sequence to make computations of the homology of coTHH(C) for certain coalgebra spectra C. We exploit the coalgebra structure in the coB¨okstedt spectral sequence to prove that the spectral sequence collapses at E2 in particular situations and give several specific examples. 1.1. Acknowledgements. The authors express their gratitude to the organizers of the Women in Topology II Workshop and the Banff International Research Station, where much of the research presented in this article was carried out. We also thank Kathryn Hess, Ben Antieau, and Paul Goerss for several helpful conversations. The second author was supported by the National Science Foundation CAREER 4 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN Grant, DMS-1149408. The third author was supported by the DNRF Niels Bohr Professorship of Lars Hesselholt. The fourth author was supported by the National Science Foundation, DMS-140648. The participation of the first author in the workshop was partially supported by the AWM’s ADVANCE grant. 2. coHochschild homology: Definitions and examples Let (D, ⊗, 1) be a symmetric monoidal category. We will suppress the associa- tivity and unitality isomorphisms that are part of this structure in our notation. The coHochschild homology of a coalgebra in D is defined as the homotopy limit of a cosimplicial object in D. In this section, we define coHochschild homology and then discuss a variety of algebraic examples.
Load more

Recommended publications
  • Commutative Hopf Algebras and Cocommutative Hopf Algebras in Positive Characteristic
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 79, 375-392 (1982) Commutative Hopf Algebras and Cocommutative Hopf Algebras in Positive Characteristic MITSUHIRO TAKEUCHI * Department of Mathematics, University of Tsukuba, Ibaraki 305, Japan Communicated by N. Jacobson Received January 1, 1981 Let A be a commutative Hopf algebra over a field k of characteristic p > 0. Let o: C + B be a surjective map of commutative algebras such that xP = 0 for any x in Ker(yl), so that the map Fc: k’@ @ C+ C, 10 a t+ lPaP factors through k’lp @ 9, yielding a map $: k”P @ B -+ C. A map of algebras f: A + B can be lifted to an algebra map f: A -+ C such that f = q 0 f if and only if Ker(F,) c Ker(Fr o (k”” @ f )). In particular, if FA is injective, any algebra map A + B can be lifted to A + C. The dual results will be given for cocommutative Hopf algebras and coalgebra maps. It is known that every reduced algebraic group over a perfect field is smooth [ 1, p. 2391. This means that if A is a finitely generated commutative Hopf algebra over a field k such that k@ A is reduced, then for any surjective map of commutative algebras p: C+ B whose kernel Ker(q) is nilpotent, any algebra mapf: A + B can be lifted to an algebra mapf: A -+ C such that v, of= f [I, Corollary 4.6, p.
    [Show full text]
  • The Formal Theory of Hopf Algebras, Part II
    The Formal Theory of Hopf Algebras Part II: The case of Hopf algebras Hans{E. Porst Department of Mathematical Sciences, University of Stellenbosch, Stellenbosch, South Africa [email protected] Abstract The category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then HopfR is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of HopfR. Moreover it is shown that, for every commutative unital ring R, the so-called \dual algebra functor" has a left adjoint and that, more generally, universal measuring coalgebras exist. MSC 2000: Primary 16T05, Secondary 18D10 Keywords: Hopf algebras, ?bialgebras, limits, colimits, free Hopf algebras, cofree Hopf algebras, Hopf envelope, universal measuring coalgebra Introduction The first monograph on Hopf algebra theory (Sweedler 1969 [27]) paid considerable attention to categorical properties of those. Somewhat surprisingly more recent successors like e.g. [11] | though using categorical language throughout | hardly touch these questions. The question we here have in mind are, e.g., Does the category of Hopf algebras have products (or, more generally, all limits) and how are they constructed? Does it have colimits? Or Do the naturally occurring functors (e.g., the embedding of the category of Hopf algebras into that of bialgebras) have adjoints? Some scattered results exist, as for example 1.
    [Show full text]
  • A Necessary and Sufficient Condition for Induced Model Structures
    A NECESSARY AND SUFFICIENT CONDITION FOR INDUCED MODEL STRUCTURES KATHRYN HESS, MAGDALENA KE¸DZIOREK, EMILY RIEHL, AND BROOKE SHIPLEY Abstract. A common technique for producing a new model category struc- ture is to lift the fibrations and weak equivalences of an existing model struc- ture along a right adjoint. Formally dual but technically much harder is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, there is a well-known necessary \acyclicity" condition. We show that for a broad class of accessible model structures | a generalization introduced here of the well-known combinato- rial model structures | this necessary condition is also sufficient in both the right-induced and left-induced contexts, and the resulting model category is again accessible. We develop new and old techniques for proving the acyclity condition and apply these observations to construct several new model struc- tures, in particular on categories of differential graded bialgebras, of differential graded comodule algebras, and of comodules over corings in both the differen- tial graded and the spectral setting. We observe moreover that (generalized) Reedy model category structures can also be understood as model categories of \bialgebras" in the sense considered here. Contents 1. Introduction 2 1.1. Acknowledgments 5 2. Accessible model categories, acyclicity, and induced model structures 5 2.1. Induced model structures and acyclicity 5 2.2. Techniques for proving acyclicity 8 2.3. Adjoint squares 11 3. Left and right induced algebraic weak factorization systems 13 3.1. Algebraic weak factorization systems and accessible model categories 14 3.2.
    [Show full text]
  • Cocommutative Coalgebras: Homotopy Theory and Koszul Duality
    Homology, Homotopy and Applications, vol. 18(2), 2016, pp.303–336 COCOMMUTATIVE COALGEBRAS: HOMOTOPY THEORY AND KOSZUL DUALITY JOSEPH CHUANG, ANDREY LAZAREV and W. H. MANNAN (communicated by J.P.C. Greenlees) Abstract We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between commu- tative and Lie algebras extends to a Quillen equivalence between cocommutative coalgebras and formal coproducts of curved Lie algebras. 1. Introduction Differential graded (dg) coalgebras arise naturally as invariants of topological spaces, for example as the normalized singular chains of a space. They also serve as representing objects for formal deformation functors [8, 13, 16] and feature promi- nently in rational homotopy theory [17, 14]. As such it is natural to ask if they may be placed in the framework of a closed model category (CMC), at least in the case of cocommutative dg coalgebras over a field of characteristic zero. The first result of this kind is due to D. Quillen [17] under a rather strong connectivity as- sumption; this assumption was subsequently removed by Hinich in [8]. The crucial difference between these two approaches is that Quillen defined weak equivalences to be quasi-isomorphisms whereas Hinich considered a finer (i.e., harder to satisfy) notion of a filtered quasi-isomorphism. A particularly nice feature of Hinich’s model is the so-called Koszul duality: it turns out to be Quillen equivalent to the category of dg Lie algebras, again without any grading restrictions.
    [Show full text]
  • Cofree Coalgebras and Differential Linear Logic
    Cofree coalgebras and differential linear logic James Clift, Daniel Murfet January 9, 2019 Abstract We prove that the semantics of intuitionistic linear logic in vector spaces which uses cofree coalgebras is also a model of differential linear logic, and that the Carte- sian closed category of cofree coalgebras is a model of the simply-typed differential lambda calculus. Contents 1 Introduction 2 2 Sweedler semantics 5 2.1 Cofreecoalgebras ................................ 5 2.2 Group-like elements are points . 13 2.3 Primitive elements are tangent vectors . 13 2.4 Definition of the Sweedler semantics . 14 2.5 Local cohomology and distributions . 17 3 Differential linear logic 18 3.1 Codereliction, cocontraction, coweakening . ... 23 arXiv:1701.01285v2 [cs.LO] 7 Jan 2019 4 Examples 25 4.1 Churchnumerals ................................ 26 4.2 Binaryintegers ................................. 27 4.3 Multiplication . 31 5 Differential lambda calculus 32 5.1 Cartesian differential categories . 36 A Kleisli categories and coalgebras 43 1 In the discrete world of computing, there is no meaningful metric in which “small” changes and “small” effects go hand in hand, and there never will be. E.W.Dijkstra, On the cruelty of really teaching computer science 1 Introduction The idea of taking derivatives of programs is an old one [34, §2] with manifestations in- cluding automatic differentiation of algorithms computing real-valued functions [8] and incremental computation [35]. However, these approaches are limited to restricted classes of computations, and it is only recently with the development of the differential λ-calculus by Ehrhard-Regnier [13] and its refinement by differential linear logic [2, 11] that deriva- tives have been defined for general higher-order programs.
    [Show full text]
  • Cofree Coalgebras Over Operads and Representative Functions
    Cofree coalgebras over operads and representative functions M. Anel September 16, 2014 Abstract We give a recursive formula to compute the cofree coalgebra P _(C) over any colored operad P in V = Set; CGHaus; (dg)Vect. The construction is closed to that of [Smith] but different. We use a more conceptual approach to simplify the proofs that P _ is the cofree P -coalgebra functor and also the comonad generating P -coalgebras. In a second part, when V = (dg)Vect is the category of vector spaces or chain com- plexes over a field, we generalize to operads the notion of representative functions of [Block-Leroux] and prove that P _(C) is simply the subobject of representative elements in the "completed P -algebra" P ^(C). This says that our recursion (as well as that of [Smith]) stops at the first step. Contents 1 Introduction 2 2 The cofree coalgebra 4 2.1 Analytic functors . .4 2.2 Einstein convention . .5 2.3 Operads and associated functors . .7 2.4 Coalgebras over an operad . 10 2.5 The comonad of coendomorphisms . 12 2.6 Lax comonads and their coalgebras . 13 2.7 The coreflection theorem . 16 3 Operadic representative functions 21 3.1 Representative functions . 23 3.2 Translations and recursive functions . 25 3.3 Cofree coalgebras and representative functions . 27 1 1 Introduction This work has two parts. In a first part we prove that coalgebras over a (colored) operad P are coalgebras over a certain comonad P _ by giving a recursive construction of P _. In a second part we prove that the recursion is unnecessary in the case where the operad is enriched over vector spaces or chain complexes.
    [Show full text]
  • Cofree Compositions of Coalgebras
    FPSAC 2011, Reykjavik, Iceland DMTCS proc. (subm.), by the authors, 1–12 Cofree compositions of coalgebras Stefan Forcey1, Aaron Lauve2†, and Frank Sottile3‡ 1Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325 USA 2Department of Mathematics, Loyola University of Chicago, Chicago, IL 60660 USA 3Department of Mathematics, Texas A&M University, College Station, TX 77843 USA Abstract. We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes. Resum´ e.´ Keywords: multiplihedron, cofree coalgebra, Hopf algebra, operad, species 1 Introduction The Hopf algebras of ordered trees (Malvenuto and Reutenauer (1995)) and of planar binary trees (Loday and Ronco (1998)) are cofree coalgebras that are connected by cellular maps from permutahedra to as- sociahedra. Related polytopes include the multiplihedra (Stasheff (1970)) and the composihedra (Forcey (2008b)), and it is natural to study what Hopf structures may be placed on these objects. The map from permutahedra to associahedra factors through the multiplihedra, and in (Forcey et al. (2010)) we used this factorization to place Hopf structures on bi-leveled trees, which correspond to vertices of multiplihedra. Multiplihedra form an operad module over associahedra.
    [Show full text]
  • A Study of Categories of Algebras and Coalgebras
    A Study of Categories of Algebras and Coalgebras Jesse Hughes May, 2001 Department of Philosophy Carnegie Mellon University Pittsburgh PA 15213 Thesis Committee Steve Awodey, Co-Chair Dana Scott, Co-Chair Jeremy Avigad Lawrence Moss, Indiana University Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the well-known dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is well-behaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem's of G. Birkhoff [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of \coequation" is introduced and several examples given.
    [Show full text]
  • Homotopy Theory of Homotopy Algebras 3
    HOMOTOPY THEORY OF HOMOTOPY ALGEBRAS BRUNO VALLETTE In memoriam JLL ABSTRACT. This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion of mor- phisms called infinity-morphisms. The method consists in using the operadic calculus to endow the category of coalgebras over the Koszul dual cooperad or the bar construction with a new type of model category structure, Quillen equivalent to that of algebras. We provide an explicit homotopy equivalence for infinity-morphisms, which gives a simple description of the homotopy category, and we endow the category of homotopy algebras with an infinity-category structure. CONTENTS Introduction 1 1. Recollections 4 2. Model category structure for coalgebras 8 3. Homotopy theory of infinity-morphisms 18 4. Homotopy algebras 22 Appendix A. Obstruction theory for infinity-morphisms 29 Appendix B. A technical lemma 30 References 32 INTRODUCTION To define derived functors in non-necessarily additive setting, D. Quillen generalized the ideas of A. Grothendieck [Gro57] and introduced the notion of model category [Qui67]. A derived functor, being de- fined up to quasi-isomorphisms, it finds its source in the homotopy category, which is the original category localized with respect to quasi-isomorphisms. (This process is the categorical analogue of the construction of the field of rational numbers, where one starts from the ring of integers and formally introduce inverses arXiv:1411.5533v3 [math.AT] 7 Feb 2016 for the non-zero numbers). For instance, Quillen homology theory for algebras of “any” type is defined by deriving the functor of indecomposables, see [LV12, Chapter 12].
    [Show full text]
  • Algebraic Deformations and Triple Cohomology
    PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 4, April 1980 ALGEBRAIC DEFORMATIONS AND TRIPLE COHOMOLOGY THOMAS F. FOX1 Abstract. The fundamental theorems of algebraic deformation theory are shown to hold in the context of enriched triple cohomology. This unifies and generalizes the classical theory. The fundamental results in algebraic deformation theory connect the low order cohomology groups of an algebra A with the existence of deformations of the algebra structure on A. The theory for associative algebras was initiated by Gerstenhaber in [6], where he also outlined techniques applicable to other special cases (see [7]). Here we give a unified treatment of the deformation theories for a broad class of algebra types using the enrichment over the category of coalgebras and triple cohomology (see [1] and [2]). Let R be a commutative ring and let (T, p, tj) be a triple on the category Mod-R. Recall that a T-algebra structure on an R-module A is given by a map a: AT-^ A satisfying the equations: (a)T • a = p- a: AT2-*A, tj • a = id: A —>A. (1) Following Gerstenhaber, we would like to investigate when a formal power series a + 2a„x" with coefficients in HomR(AT, A) determines a formal T-algebra structure on A (or a T-algebra structure on ^4[[x]], see [6]). Direct use of the conditions (1) immediately brings up the problem of the nonadditivity of most triples of interest, e.g. the tensor-algebra triple. To circumvent this problem we must use an additive enrichment of T over the category of /?-coalgebras.
    [Show full text]
  • On Sweedler's Cofree Cocommutative Coalgebra
    On Sweedler’s cofree cocommutative coalgebra Daniel Murfet August 22, 2018 Abstract We give a direct proof of a result of Sweedler describing the cofree cocommutative coalgebra over a vector space, and use our approach to give an explicit construc- tion of liftings of maps into this universal coalgebra. The basic ingredients in our approach are local cohomology and residues. 1 Introduction Let k be an algebraically closed field of characteristic zero. Given a vector space V there is a universal cocommutative coalgebra !V mapping to V which is sometimes called the cofree cocommutative coalgebra generated by V . This is a classical construction going back to Sweedler, and an explicit description follows from his results in [24]: for V finite- dimensional the universal cocommutative coalgebra may be presented as a coproduct !V = SymP (V ) (1.1) P ∈V M where SymP (V ) = Sym(V ) is the symmetric coalgebra. The universal map d :!V −→ V ⊗i is defined on components vi ∈ V by the formula arXiv:1406.5749v2 [math.RA] 4 Jan 2017 d|SymP (V ) : SymP (V ) −→ V, (v0, v1, v2,...) 7−→ v0P + v1 . While not written explicitly in [24] this is a straightforward exercise using the structure theory of coalgebras developed there. We do this exercise in Appendix B. However, since Sweedler’s description (1.1) of !V seems not to be well-known despite the extensive discussion of cofree cocommutative coalgebras in models of logic [4, 10, 18] and elsewhere it seems worthwhile to give a more direct proof. Our main new result is an explicit description of the lifting of a linear map !W −→ V to a morphism of coalgebras !W −→ !V (Theorem 2.22).
    [Show full text]
  • Cofree Compositions of Coalgebras
    FPSAC 2011, Reykjavik, Iceland DMTCS proc. (subm.), by the authors, 1–12 Cofree compositions of coalgebras Stefan Forcey1, Aaron Lauve2†, and Frank Sottile3‡ 1Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325 USA 2Department of Mathematics, Loyola University of Chicago, Chicago, IL 60660 USA 3Department of Mathematics, Texas A&M University, College Station, TX 77843 USA Abstract. We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude with examples of these structures, where the factor coalgebras have bases indexed by the vertices of multiplihedra, composihedra, and hypercubes. Resum´ e.´ Nous d´eveloppons la notion de la composition de deux coalg`ebres, qui apparaˆıt naturellement dans la th´eorie de cat´egorie plus ´elev´ees et de la th´eorie des esp`eces. Nous montrons que la composition de deux coalg`ebres colibre est colibre et nous donnons des conditions qui impliquent que la composition est une alg`ebre de Hopf unilat´erale. Ces conditions sont valables quand une des coalg`ebres est une op´erade de Hopf gradu´ee D et l’autre est une coalg`ebre gradu´ee connexe avec un morphisme coalg`ebre `a D. Nous concluons avec des exemples de ces structures, o`ules coalg`ebres compos´ees ont des bases index´ees par les sommets de multiplih´edra, composih´edra, et hypercubes.
    [Show full text]